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Polar coordinates

Polar coordinate systems are coordinate systems in which a point is identified by a distance from some fixed feature in space and one or more subtended angles.

The principal types of polar coordinate systems are listed below.

Circular Polar Coordinates

A two-dimensional coordinate system, defined by an origin, O, and a semi-infinite line L leading from this point. L is also called the polar axis. In terms of the Cartesian coordinate system, one usually picks O to be the origin (0,0) and L to be the positive x-axis (the right half of the x-axis).

A point P is then located by its distance from the origin and the angle between line L and OP, measured anti-clockwise. The co-ordinates are typically denoted r and θ respectively: the point P is then (r, θ).

Cylindrical Polar Coordinates

(Also see cylindrical coordinate system)

A three-dimensional system which essentially extends circular polar coordinates by adding a third co-ordinate (usually denoted h) which measures the height of a point above the plane.

A point P is given as (r, θ, h). In terms of the Cartesian system:

  • r is the distance from O to P', the orthogonal projection of the point P onto the XY plane. This is the same as the distance of P to the z-axis.
  • θ is the angle between the positive x-axis and the line OP', measured anti-clockwise.
  • h is the same as z.
Some mathematicians indeed use (r, θ, z).

Spherical Polar Coordinates

(Also see spherical coordinates.)

This system is another way of extending the circular polar system to three dimensions, defined by a line in a plane and a line perpendicular to the plane. (The x-axis in the XY plane and the z-axis.)

For a point P, the distance co-ordinate is the distance OP, not the projection. It is sometimes notated r but often ρ (Greek letter rho) is used to emphasise that it is in general different to the r of cylindrical co-ordinates.

The remaining two co-ordinates are both angles: θ is the anti-clockwise between the x-axis and the line OP', where P' is the projection of P in the XY-axis. The angle φ, measures the angle between the vertical line and the line OP.

In this system, a point is then given as (ρ, φ, θ).

Note that r = ρ only in the XY plane, that is when φ= π/2 or h=0.

See also:

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